Following the global strategy introduced recently by Bona, Lannes and Saut in Ref. 7, we derive here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, the presence of surface tension and with uneven bottoms. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on ℝ^d, d = 1, 2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators and the surface tension term with respect to suitable small parameters that depend variously on the amplitude, wavelengths and depth ratio of the two layers. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established. © 2009 World Scientific Publishing Company.

Anh C.T.

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Từ khóa : Asymptotic model; Asymptotic systems; Bottom surfaces; Bottom topography; Depth ratio; Dirichlet-Neumann operator; Evolution equations; Global strategies; Internal waves; Large class; Nonlocal operator; Scaling regimes; Two layers; Uneven bottom; Asymptotic analysis; Capillarity; Differential equations; Euler equations; Surface chemistry; Surface properties; Surface tension; Surface topography; Wetting; Mathematical operators